The repeating decimals $0.abab\overline{ab}$ and $0.abcabc\overline{abc}$ satisfy
\[0.abab\overline{ab}+0.abcabc\overline{abc}=\frac{33}{37},\]
where $a$, $b$, and $c$ are (not necessarily distinct) digits. Find the three digit number $abc$.

Answer: Notice repeating decimals can be written as the following:
$0.\overline{ab}=\frac{10a+b}{99}$
$0.\overline{abc}=\frac{100a+10b+c}{999}$
where a,b,c are the digits. Now we plug this back into the original fraction:
$\frac{10a+b}{99}+\frac{100a+10b+c}{999}=\frac{33}{37}$
Multiply both sides by $999*99.$ This helps simplify the right side as well because $999=111*9=37*3*9$:
$9990a+999b+9900a+990b+99c=33/37*37*3*9*99=33*3*9*99$
Dividing both sides by $9$ and simplifying gives:
$2210a+221b+11c=99^2=9801$
At this point, seeing the $221$ factor common to both a and b is crucial to simplify. This is because taking $mod 221$ to both sides results in:
$2210a+221b+11c  \equiv 9801 \mod 221 \iff 11c  \equiv 77 \mod 221$
Notice that we arrived to the result $9801 \equiv 77 \mod 221$ by simply dividing $9801$ by $221$ and seeing $9801=44*221+77.$ Okay, now it's pretty clear to divide both sides by $11$ in the modular equation but we have to worry about $221$ being multiple of $11.$ Well, $220$ is a multiple of $11$ so clearly, $221$ couldn't be. Also, $221=13*17.$ Now finally we simplify and get:
$c \equiv 7 \mod 221$
But we know $c$ is between $0$ and $9$ because it is a digit, so $c$ must be $7.$ Now it is straightforward from here to find $a$ and $b$:
$2210a+221b+11(7)=9801 \iff 221(10a+b)=9724 \iff 10a+b=44$
and since a and b are both between $0$ and $9$, we have $a=b=4$. Finally we have the $3$ digit integer $\boxed{447}$.